JHEP07(2008)103 July 9, 2008 July 24, 2008 April 15, 2008 Ries

Published by Institute of Physics Publishing for SISSA Received: April 15, 2008 Accepted: July 9, 2008 Published: July 24, 2008 Polyakov formulas for GJMS operators from AdS/CFT JHEP07(2008)103 Danilo E. D´ıaz Humboldt-Universität zu Berlin, Institut für Physik, Newtonstr.15, D-12489 Berlin, Germany E-mail: [email protected] Abstract: We argue that the AdS/CFT calculational prescription for double-trace deformations leads to a holographic derivation of the conformal anomaly, and its conformal primitive, associated to the whole family of conformally covariant powers of the Laplacian (GJMS operators) at the conformal boundary. The bulk side involves a quantum 1-loop correction to the SUGRA action and the boundary counterpart accounts for a sub-leading term in the large-N limit. The sequence of GJMS conformal Laplacians shows up in the two-point function of the CFT operator dual to a bulk scalar field at certain values of its scaling dimension. The restriction to conformally flat boundary metrics reduces the bulk computation to that of volume renormalization which renders the universal type A anomaly. In this way, we directly connect two chief roles of the Q-curvature: the main term in Polyakov formulas on one hand, and its relation to the Poincare metrics of the Fefferman-Graham construction, on the other hand. We find agreement with previously conjectured patterns including a generic and simple formula for the type A anomaly coefficient that matches all reported values in the literature concerning GJMS operators, to our knowledge. Keywords: AdS-CFT Correspondence, Anomalies in Field and String Theories. Contents 1. Introduction 1 2. The physical motivation: AdS/CFT correspondence 5 2.1 The generalized prescription for double-trace deformations 5 2.2 Bulk one-loop effective actions 6 JHEP07(2008)103 2.3 Boundary partition function 6 2.4 The functional determinants 7 3. Digression on regularizations 8 4. The general case of Poincare metrics 11 5. Conformally flat class and volume renormalization 12 5.1 The infinitesimal variation: conformal anomaly 13 5.2 Type A holographic anomaly 13 5.3 Conformal primitive: Polyakov formulas 14 6. Comparison to “experiment” 14 6.1 Further data for the conformal Laplacian 16 7. Conclusion 17 A. GJMS operators and Q-curvature 19 B. Q-curvature and volume renormalization 20 C. Rigid case in dimensional regularization 22 1. Introduction Conformally covariant differential operators have been the subject of a continuous inter- play between physics and mathematics ever since the discovery of conformal invariance of Maxwell’s equations by Cunningham [1] and Bateman [2] in the early part of last century. To this early list belongs the Dirac operator, after Pauli’s proof of the conformal invariance of the massless Dirac equation [3], as well as the conformal wave operator, both in curved spacetime. The Riemannian variant of the later, the conformal Laplacian, is best known to mathematicians for its role in the Yamabe problem of prescribing scalar curvature on a Riemannian manifold [4]. – 1 – In the early 80’s a fourth-order conformal covariant was found by Paneitz [5], and independently by Eastwood and Singer [6], in relation with gauge fixing Maxwell equations respecting conformal symmetry. It was rediscovered by Riegert [7] while pursuing a different goal, namely a four-dimensional analog of Polyakov formula [8] for the conformal (or trace, or Weyl) anomaly, i.e. a non-local covariant action whose conformal variation leads to the general form of the anomaly. Graham, Jenne, Mason and Sparling [9] further showed that the conformal Laplacian and the Paneitz operator generalize to a family of k conformally covariant differential operators P2k of even order 2k, with leading term ∆ , whenever the dimension d of the manifold is odd or d 2k. These ‘conformally covariant ≥ JHEP07(2008)103 powers of the Laplacian’ (“GJMS” operators in what follows) P2k were obtained using the Fefferman-Graham ambient metric [10], a chief tool for the systematic construction of conformal invariants.1 The feature of the GJMS operators that we will treat in this paper is the conformal variation of their functional determinant2 encoded in a (generalized) Polyakov formula. In other words, we focus on the conformal anomaly, and its conformal primitive, associated to this family of operators. These formulas can be worked out case by case in low dimensions from heat kernel coefficients whose complexity grows significantly with the dimension of the compact manifold . However, Branson [13] succeeded in finding a pattern in terms M of the Q-curvature, in the conformally flat case, to rewrite ‘more invariantly’ the quotient of functional determinants of a conformally covariant operator A (or a power thereof and with suitable positive ellipticity properties) at conformally related metrics g = e 2wg in any even dimension d: det A b log = c w Q dv + Q dv + F dv F dv + (global term) (1.1a) − det A d gˆ d g gˆ − g ZM ZM b 1 =2c w Qb d + Pdw dvg + F dvb gˆ F dvg +(global term) . (1.1b) M 2 M − Z Z Here F stands for a local curvature invariant and theb global term is related to the null space (kernel) of the operator; both vary depending on what A is, whereas the universal part is captured by the Q-curvature term. Away from conformal flatness, the pattern is preserved in d = 2, 4, 6 and conjectured to hold in all even dimension. A related conjecture by Deser and Schwimmer [14] expresses the infinitesimal variation (anomaly) as a combination of the Euler density (or Pfaffian), a local conformal invariant and a total derivative;3 but it can be rephrased in terms of the Q-curvature instead of the Pfaffian. It was in this context that Branson first defined the Q-curvature, in general even dimension, from the zeroth order (in derivatives) term of the GJMS operators via analytic continuation in the dimension. The linear transformation law under conformal rescaling g = e 2wg of the Q-curvature generalizes that of the scalar curvature in two dimensions, edw Q = Q + P w , (1.2) b d d d 1 In a physical setting, see the recent work [11,b 12] for an alternative route. 2In the mathematical literature, the zeta-regularized functional determinant is usually meant. 3See [15] and [16] for independent proofs. The restriction to the conformally flat class, where the Weyl tensor vanishes, was anticipated in [17]. – 2 – and integrates to a multiple of the Euler characteristic on conformally flat manifolds. Another context in which the Q-curvature plays a central role arises in the volume renormalization of conformally compact asymptotically Einstein manifolds [18], i.e. manifolds whose filling metric is the Poincare metric of the Fefferman-Graham construction. The renewed interest in the seminal work of Fefferman and Graham [10], as an outgrowth of the relation between the geometry of hyperbolic space (Lobachevsky space) and conformal geometry on the sphere at the conformal infinity, has been triggered by the AdS/CFT Correspondence in physics [19 – 21]. The reconstruction of a bulk metric associated to a given conformal structure at the conformal infinity [22] and the subsequent evaluation of the Einstein action with a negative cosmological term becomes the geometrical task in the JHEP07(2008)103 limit in which the effective description of string theory is featured by the classical super- gravity approximation. When the bulk metric is Einstein, the Lagrangian factorizes and the challenge consists in the regularization of the infinite (infrared divergent) volume. A key feature of the AdS/CFT duality is the fact that infrared divergences in the bulk are related to ultraviolet ones on the boundary theory, the so called IR-UV connection [23]. A further elaboration thereof leads to the mapping of the conformal anomaly of the gauge theory on the boundary, at large N (rank of the gauge group) and large ’t Hooft coupling, to the failure of the renormalized bulk action to be independent of the conformal representative of the metric at the boundary. This analysis was thoroughly carried out by Henningson and Skenderis [24]. The holographic anomaly associated to the volume is given by the coefficient v(d) of the volume expansion and its integral gives the coefficient of the log-divergent term L in the volume asymptotics. The Q-curvature enters here after the observation by Graham and Zworski [25] that the “integrated anomaly” is also proportional to the integral of L the Q-curvature. A further refinement by Graham and Juhl [26] results in a holographic formula for the Q-curvature in terms of the coefficients of the volume expansion. These later developments involve scattering theory for Poincare metrics associated to the conformal structure as an alternative route to GJMS operators and to Q-curvature [25, 27]. They are influenced by, and at the same time generalize, results originally discussed in the physical context by Witten [21] for the “rigid” case of hyperbolic space. They are also closely related to the AdS/CFT computation of matter conformal anomalies which show up in the trace of the holographic energy-momentum tensor, in addition to the purely geometric terms involving curvature invariants of the boundary metric, for certain non- trivial profiles of the matter fields. Here, the powers of the Laplacian that arise in the rigid case [28 – 30] naturally generalize to the GJMS operators, as outlined in [29, 30]. In particular, the GJMS Laplacians show up as residues of the scattering operator SM (λ) at the poles λ = d/2+ k, k N. The “rigid” version of the scattering operator S (λ) is the ∈ M two-point function O O of a CFT operator O of conformal dimension λ on Rd or Sd h λ λi λ as conformal boundary of the half-space model or of the ball model of the hyperbolic space Hd+1, respectively.

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